Principal component analysis

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# Principal component analysis

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# Why reduce dimensions?

High dimensional data is difficult to analyse:
- many variables (p)
- difficult to visualise and intuit
- sometimes no clear response variable.

Reducing dimensionality can retain (most of) the original information
and make the data easier to understand and work with.

Principal Component Analysis is one such method for identifying the main
sources of variation in a dataset.

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# What is PCA (informally)?

PCA finds linear combinations of the input features that explain a large amount
of the variation in the data, combining them into new features called
"principal components" (PC).

The first PC has largest explanatory power.

Each of these PCs is independent from the previous one.

The contribution of each feature to each PC is measured by its loading.

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# What is PCA (formally)?

The first principal component ($Z_1$) is calculated using the equation:

$$  
  Z_1 = w_{11}X_1 + w_{21}X_2 +....+ w_{p1}X_p
$$

`$X_1, ..., X_p$` represents variables in the original dataset and `$w_{11}, ..., w_p$`
represent principal component loadings, which can be thought of as the degree to
which each variable contributes to the calculation of the principal component.

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# PCA finds the main sources of variation in the data